According to anonymous sources who are quoted by royal reporter Kate Mansey in The Times, and who appear to be close (or claiming to be) to the Duke and Duchess of Sussex, their two children began to be known by the names Archie Sussex and Lilibet Sussex when they became Prince Archie of Sussex and Princess Lilibet of Sussex.
Quoting from the article:
Rather than being known as the Mountbatten-Windsors, Prince Archie and Princess Lilibet had been known as Archie Sussex and Lilibet Sussex since the coronation, a source said — in the same way that Harry was known as Captain Wales when he was in the army.
https://www.thetimes.co.uk/article/...e-lilibet-children-name-royal-title-cnvf7d9jf
https://archive.ph/1l2I9
For those unfamiliar with it, this is longstanding convention for British princes and princesses and British peers and peeresses, all of whom have legal surnames in theory but, by tradition, do not use them.
Outside of legal settings (where, for obvious reasons, legal names are used), princes(ses) and peer(esse)s who must or choose to use a surname in certain contexts (such as schooling or business) typically assume the territorial designations from their titles as their unofficial "surname".
For example, the late Earl of Snowdon used Antony Snowdon as his professional name, and his son, then Viscount Linley, began his career as David Linley. The then Prince William and Prince Henry of Wales and the then Princess Beatrice and Princess Eugenie of York held jobs as William Wales, Harry Wales, Beatrice York, and Eugenie York. (Prince Edward, who was The Prince Edward and had no territorial designation when he embarked on his professional career, used Edward Windsor until he became Earl of Wessex and then switched to Edward Wessex.)
A further title-related comment from an unidentified "source ":
Their foundation and production company are called Archewell, inspired by their four-year-old son’s name. However, the new website [sussex.com] is thought to be intended to be inclusive of both children.
After publishing his autobiography Spare, which detailed a childhood of being treated as a less-important “spare” to the “heir”, Prince William, the Sussexes are said to be keen not to discriminate against their own second-born, two-year-old Lilibet.
A source said: “The reality behind the new site is very simple — it’s a hub for the work the Sussexes do and it reflects the fact the family have, since the King’s coronation, the same surname for the first time. That’s a big deal for any family. It represents their unification and it’s a proud moment.”
It would be interesting to know how the source who claims "After publishing his autobiography Spare, which detailed a childhood of being treated as a less-important “spare” to the “heir”, Prince William, the Sussexes are said to be keen not to discriminate against their own second-born, two-year-old Lilibet" would explain that:
1) the Sussexes accepted a hereditary peerage with a standard male-only succession, which they were well aware would discriminate not only against any second-born son, but against any daughter, even if firstborn? Lilibet cannot inherit the dukedom whereas Archie can, and that would still be the case even if Lilibet were the elder child.
2) the Sussexes accepted a princely title for their children which their daughter Lilibet will lose part of (specifically, she will lose the "of Sussex") if and when she marries and be unable to share with her spouse, while their son Archie will keep the full title for life and share it with any female spouse he might have, who will become Princess Archie of Sussex?